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By Root 548 0

races down Main Street swigging from a half-empty $2 bottle of Boone’s Farm apple wine)? Since the company has no data on you—no idea of the “position of the first ball”—it might assign you an equal prior probability of being in either group, or it might use what it knows about the general population of new drivers and start you off by guessing that the chances you are a high risk are, say, 1 in 3. In that case the company would model you as a hybrid—one-third high risk and two-thirds low risk—and charge you one-third the price it charges high-risk drivers plus two-thirds the price it charges low-risk drivers. Then, after a year of observation—that is, after one of Bayes’s second balls has been thrown—the company can employ the new datum to reevaluate its model, adjust the one-third and two-third proportions it previously assigned, and recalculate what it ought to charge. If you have had no accidents, the proportion of low risk and low price it assigns you will increase; if you have had two accidents, it will decrease. The precise size of the adjustment is given by Bayes’s theory. In the same manner the insurance company can periodically adjust its assessments in later years to reflect the fact that you were accident-free or that you twice had an accident while driving the wrong way down a one-way street, holding a cell phone with your left hand and a doughnut with your right. That is why insurance companies can give out “good driver” discounts: the absence of accidents elevates the posterior probability that a driver belongs in a low-risk group.

Obviously many of the details of Bayes’s theory are rather complex. But as I mentioned when I analyzed the two-daughter problem, the key to his approach is to use new information to prune the sample space and adjust probabilities accordingly. In the two-daughter problem the sample space was initially (boy, boy), (boy, girl), (girl, boy), and (girl, girl) but reduces to (boy, girl), (girl, boy), and (girl, girl) if you learn that one of the children is a girl, making the chances of a two-girl family 1 in 3. Let’s apply that same simple strategy to see what happens if you learn that one of the children is a girl named Florida.

In the girl-named-Florida problem our information concerns not just the gender of the children, but also, for the girls, the name. Since our original sample space should be a list of all the possibilities, in this case it is a list of both gender and name. Denoting “girl-named-Florida” by girl-F and “girl-not-named-Florida” by girl-NF, we write the sample space this way: (boy, boy), (boy, girl-F), (boy, girl-NF), (girl-F, boy), (girl-NF, boy), (girl-NF, girl-F), (girl-F, girl-NF), (girl-NF, girl-NF), and (girl-F, girl-F).

Now, the pruning. Since we know that one of the children is a girl named Florida, we can reduce the sample space to (boy, girl-F), (girl-F, boy), (girl-NF, girl-F), (girl-F, girl-NF), and (girl-F, girl-F). That brings us to another way in which this problem differs from the two-daughter problem. Here, because it is not equally probable that a girl’s name is or is not Florida, not all the elements of the sample space are equally probable.

In 1935, the last year for which the Social Security Administration provided statistics on the name, about 1 in 30,000 girls were christened Florida.5 Since the name has been dying out, for the sake of argument let’s say that today the probability of a girl’s being named Florida is 1 in 1 million. That means that if we learn that a particular girl’s name is not Florida, it’s no big deal, but if we learn that a particular girl’s name is Florida, in a sense we’ve hit the jackpot. The chances of both girls’ being named Florida (even if we ignore the fact that parents tend to shy away from giving their children identical names) are therefore so small we are justified in ignoring that possibility. That leaves us with just (boy, girl-F), (girl-F, boy), (girl-NF, girl-F), and (girl-F, girl-NF), which are, to a very good approximation, equally likely.

Since 2 of the 4, or half, of the elements in the sample space